98,167 research outputs found

    Weighted Well-Covered Claw-Free Graphs

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    A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input claw-free graph G, we present an O(n^6)algortihm, whose input is a claw-free graph G, and output is the vector space of weight functions w, for which G is w-well-covered. A graph G is equimatchable if all its maximal matchings are of the same cardinality. Assume that a weight function w is defined on the edges of G. Then G is w-equimatchable if all its maximal matchings are of the same weight. For every graph G, the set of weight functions w such that G is w-equimatchable is a vector space. We present an O(m*n^4 + n^5*log(n)) algorithm which receives an input graph G, and outputs the vector space of weight functions w such that G is w-equimatchable.Comment: 14 pages, 1 figur

    Determining the Solution Space of Vertex-Cover by Interactions and Backbones

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    To solve the combinatorial optimization problems especially the minimal Vertex-cover problem with high efficiency, is a significant task in theoretical computer science and many other subjects. Aiming at detecting the solution space of Vertex-cover, a new structure named interaction between nodes is defined and discovered for random graph, which results in the emergence of the frustration and long-range correlation phenomenon. Based on the backbones and interactions with a node adding process, we propose an Interaction and Backbone Evolution Algorithm to achieve the reduced solution graph, which has a direct correspondence to the solution space of Vertex-cover. By this algorithm, the whole solution space can be obtained strictly when there is no leaf-removal core on the graph and the odd cycles of unfrozen nodes bring great obstacles to its efficiency. Besides, this algorithm possesses favorable exactness and has good performance on random instances even with high average degrees. The interaction with the algorithm provides a new viewpoint to solve Vertex-cover, which will have a wide range of applications to different types of graphs, better usage of which can lower the computational complexity for solving Vertex-cover

    Local colourings and monochromatic partitions in complete bipartite graphs

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    We show that for any 22-local colouring of the edges of the balanced complete bipartite graph Kn,nK_{n,n}, its vertices can be covered with at most~33 disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite rr-locally coloured graph with O(r2)O(r^2) disjoint monochromatic cycles.\\ We also determine the 22-local bipartite Ramsey number of a path almost exactly: Every 22-local colouring of the edges of Kn,nK_{n,n} contains a monochromatic path on nn vertices.Comment: 18 page
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